What is the importance of axis of symmetry in determining the centre of gravity?
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Answer:
You can use symmetry to determine center of gravity when certain conditions are met:
The material is isotropic. That is, the density of the material changes the same in one direction the same as any other.
The material has a symmetrical distribution of mass. Note that this is not the same thing as having uniform density, although a uniform density will satisfy this condition.
The symmetry must exist around the center of mass. You can have a symmetrical distribution which is concentrated near a boundary of your system (thinking of a large sphere of gas in a gravitational field in this case.)
The particular volume of material being considered is symmetrical.
Those conditions must be met to use symmetry to find center of gravity. A violation of any of these will make a symmetry argument that isn't true in all cases. These conditions are for a center of gravity which is internal to a system, of course. If it is external, say a pair of spherical bodies orbiting a common center of gravity, then the symmetrical distribution of mass must be true for the system itself.
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